Optical measuring system

ABSTRACT

Apparatus and methods are described for of measuring amplitude and phase variations in a spatially coherent beam of light. A beam of coherent light is made incident upon a spatial array of phase modulating elements displaying a pixellated first phase distribution. In a measuring region of said spatial array, the phase distribution is changed to a new value while retaining the first phase distribution outside the measuring region, for example by flashing a single pixel. The change in intensity resulting from the change in phase distribution is then determined.

CROSS REFERENCE

This application is a PCT National Phase Application the claims priorityfrom PCT/GB2004/005182 having an International filing date of 9 Dec.2004 which in turn claims priority from UK patent applicationGB0328904.8 filed 12 Dec. 2003.

The present invention relates to a method of measuring amplitude andphase variations within a spatially coherent beam of light, to a methodof characterising a spatially coherent beam of light, to apparatus formeasuring amplitude and phase variations in a spatially coherent beam oflight, and to apparatus for characterising a spatially coherent beam oflight.

In many optical systems throughput efficiency, crosstalk and noise leveldepend on the alignment of the various components with respect to oneanother and to the optical beams passing through. An important problemduring assembly of many free space optical systems is that the beam isnot visible and its properties cannot be measured as it is incident onthe various components in the system, except where the component happensto be a detector array. Even with a detector array measurements arerestricted to the intensity when it is the phase distribution that wouldbe more usefully measured.

Usually measurements are confined to overall coupling efficiency intothe intended output or some intermediate output.

The more components there are to adjust the more variables there are tooptimise, and working “blind” the whole procedure can be very timeconsuming and expensive. A further problem is that the beams in a realoptical system, although maybe Gaussian-like in theory, often havesidelobes due to aberrations, especially when lenses are misaligned. Asearch for a global optimum can lead to an apparent optimum alignmentthat instead results in a sidelobe being coupled into the output, ratherthan the main peak. The existence of such subsidiary maxima maims itmore difficult to optimise the alignment.

Often the components themselves are not ideal. For example lenses have atolerance in dimensions and focal length that makes pick and placeassembly inappropriate, while angle-polished fibres have a tolerance inthe polish angle. As a result the relative longitudinal spacing,transverse offset and tilt of an optical fibre and lens may need to beadjusted to suit the properties of that lens and fibre. The relativeorientation of the lens-fibre assembly may need to be adapted to therest of the system.

Liquid Crystal over Silicon spatial light modulators (LCOS SLM) aredevices having an array of elements each capable of applying acontrollable phase change to light incident upon the element. LCOS SLMsmay be used for applying phase modulation to incident light beams, andmay be one dimensional or often 2 dimensional LCOS SLMs may be used tocarry out many optical processing functions such as correlation,monitoring a beam by tapping off a small fraction of the incidentenergy, routing a beam, changing a beam focus, aberration correction,changing a beam shape or changing the power carried by a beam. Eachelement of an LCOS SLM is sometimes referred to as a pixel, even thoughit does not carry picture information; in this document the term “pixel”signifies phase modulating element.

The dimensions of the pixels of the SLM may be selected according to theapplication to which the SLM is to be put when in use in the opticalsystem (as opposed to when calibration/set-up/characterisation is takingplace). In a wavelength routing/beam steering application, the pixelmight be between 5 um and 15 um, and in this case beams might have aspot size of a few 10's of ums (e.g. 50) to say 250 um.

In an embodiment, the size of the array per beam has a width of around 3times the spot size, and height a little smaller. Typically at least 100pixels are needed for hologram display.

Where low loss is important, for example where the SLM is to function asa controllable alignment mirror, the pixel size may be selected to bebetween 20 um sq and 40 um sq. Here a beam spot radius could be 1 or 2mm.

A number of known devices and methods for characterising wavefrontsexist. Some of these require intensive computation and/or twodimensional detector devices capable in themselves of simultaneouslydetermining the spatial variation in detected light.

The present invention is advantageous in having embodiments that arecapable of using only a point detector, rather than one which providesdata indicative of the spatial distribution of light, and embodimentsthat use binary SLMs.

Embodiments of the invention are aimed at providing an ability to assistwith the alignment problems mentioned above by enabling a more completecharacterisation of beams in an optical system. The invention is howevernot restricted to this area.

According to one aspect of the invention there is provided a method ofmeasuring amplitude and phase variations in a spatially coherent beam oflight comprising causing the beam to be incident upon a spatial arraydisplaying a pixellated first phase distribution, in a measuring regionof said spatial array causing the phase distribution to assume a newvalue while retaining the first phase distribution outside the measuringregion, in the Fourier plane determining the change in intensityresulting from the change in phase distribution.

According to another aspect of the invention there is provided a methodof characterising a spatially coherent beam of light, comprisingdisposing a LCOS SLM in the path of the beam; causing the LCOS SLM todisplay a first hologram pattern; at a location in said beam where theamplitude and phase of the beam are to be characterised, changing thehologram pattern to a second hologram pattern; and measuring the effectof said change by measuring an intensity.

In an embodiment, the output from the SLM is measured in the Fourierplane to detect the Fourier output.

In an embodiment, the method comprises measuring the intensity in aregion of the Fourier plane where F₀(x,y) is very weak but g(x,y) isrelatively stronger, varying the position on the SLM where theperturbation is applied to form a set of measurements of(g(x,y)f(u₀,v₀))² taking the square root of these measurements to derivevalues for the relative field amplitude at these positions.

In an embodiment, the method comprises stepping through a sequence ofphase distributions.

In an embodiment, the method comprises varying the phase shift in arespective single pixel.

According to another aspect of the invention there is provided apparatusfor measuring amplitude and phase variations in a spatially coherentbeam of light, the apparatus comprising a pixellated spatial array, eachpixel being controllable to apply any of plural phase shifts to inputlight, whereby the array displays a desired distribution of phasemodulation, means for causing the array to display a first selecteddistribution of phase modulation; means for changing the firstdistribution in a measuring region of said spatial array to assume a newdistribution while retaining the first phase distribution outside themeasuring region, means disposed in the Fourier plane for determining achange in intensity of light resulting from the change in phasedistribution.

According to a further aspect of the invention there is providedapparatus for characterising a spatially coherent beam of light,comprising a LCOS SLM arranged so that a said beam of light can beincident upon it; means for causing the LCOS SLM to display a firsthologram pattern; means for changing the hologram pattern to a secondhologram pattern at a location in said beam where the amplitude andphase of the beam are to be characterised; and means for measuring anintensity of light to determine the effect of said change of hologrampattern.

In an embodiment, the means for measuring is disposed in theFourier-plane to detect the Fourier output.

In an embodiment, the apparatus further comprises a lens for providingthe Fourier output.

In an embodiment, the apparatus further comprises a mirror for providingthe Fourier output.

An example embodiment of the invention will now be described withreference to the following drawings, in which:

FIG. 1 shows a diagram of a set up of apparatus embodying the invention;

FIG. 2 shows a diagram of a second apparatus embodying the invention;

FIG. 3 shows a “flashing pixel”;

FIGS. 4, 5 and 6 show embodiments of apparatus for aligning an SLM orits pixel assignment with a set of one or more beams;

FIG. 7 shows an embodiment of a device used for calibrating an SLM;

FIG. 8 shows a first device using wavefront sensing to monitor a beam;

FIG. 9 shows a second device using wavefront sensing to monitor a beam;

FIG. 10 shows a third device using wavefront sensing to monitor a beam;

FIG. 11 is a diagram showing one technique for increasing a differencesignal; and

FIG. 12 shows a device having free space taps.

Referring to FIG. 1, an input fibre 1 is disposed at the focal point ofa lens 2, and an LCOS SLM 3 is disposed in the opposite focal plane.Light from the fibre 1 passes to the lens 2 and onto the SLM 3, whichdisplays a hologram pattern on its pixels so as to cause reflected light12 to be deviated by an angle θ. A photodiode 4 is shown at the positionwhere this light 12 is focussed by the lens 2. In practice the systemmay include a beam splitter to allow the disposition of the photodiode4.

Referring to FIG. 2, an arrangement is shown with a 1-D reflective phasemodulating FLC SLM with 540 pixels. The arrangement is a 2-f system witha Fourier lens, and further comprises a silica-on-silicon waveguidearray allowing light to be input and extracted.

A 2-D SLM may be used instead of a 1-D SLM. Use of a reflective SLM isnot fundamental to the invention. Transmissive LCOS SLMs may howeverimpart phase variations due to varying silicon thicknesses across theSLM.

Let (u,v) be the co-ordinate system at the SLM and let the spatiallycoherent incident beam be f(u,v) exp i φ(iv) where f(u,v) describes theamplitude and φ(u,v) describes the phase. As an example the incidentfield could be an off-axis normally incident defocused Gaussian beam.For this example, the incident-beam may be described by equation (1):

$\begin{matrix}{{{f\left( {u,v} \right)} = {\exp - \left\{ \frac{\left( {u - u_{INC}} \right)^{2} + \left( {v - v_{INC}} \right)^{2}}{\omega^{2}} \right\}}}{{\exp\;{\mathbb{i}}\;{\phi\left( {u,v} \right)}} = {\exp - {{\mathbb{i}}\; k\left\{ \frac{\left( {u - u_{INC}} \right)^{2} + \left( {v - v_{INC}} \right)^{2}}{2R} \right\}}}}} & (1)\end{matrix}$where u_(INC) and v_(INC) are the co-ordinates at the centre of theGaussian beam and R is the radius of curvature.

Consider the SLM to apply a known hologram pattern H₀(u,v) to theincident beam. The hologram may be selected according to the position ofthe detector so as to steer the beam to the detector, rather thanconstraining the detector to be at a specific location or locations. Ingeneral H₀ will be a complex function describing phase and/or amplitudemodulation. In the general case the output field from the SLM isH₀(u,v)f(u,v) exp i φ(u,v).

Measure the output from the SLM in the Fourier plane, using a suitablypositioned lens or lenses and one or more optical receiving devices todetect the Fourier output at one or more positions. Suitable opticalreceiving devices would be a photodiode, an optical fibre coupled to aphotodiode or a waveguide coupled to a fibre that is itself coupled to aphotodiode. Let (x,y) be the co-ordinate system at the Fourier plane.The origin for x and y is the position where the lens optical axisintersects the Fourier plane. Let the Fourier transform of H₀(u,v)f(u,v)exp i φ(u,v) be F₀(x,y) exp i θ(x,y) where F₀(x,y) describes theamplitude and θ(x,y) describes the phase. Hence the measured intensityis proportional to the term F₀ ²(x,y).

Now change the hologram pattern in a known way at the position (u₀,v₀)where it is required to characterise the beam phase and amplitude suchthat the hologram pattern H(u,v) becomes that shown in equation (2), asfollows:H(u,v)=H ₀(u,v)+H ₁(u,v) at (u,v) close to (u ₀ ,v ₀)  (2)H(u,v)=H ₀(u,v) elsewhere.

Hence in a known neighbourhood of the point (u₀,v₀) there is aperturbation in the hologram pattern. Therefore there is also aperturbation in the output field from the SLM, given by H₁(u,v)f(u,v)exp i φ(u,v). The incident field amplitude and phase may be approximatedto be uniform across the perturbation region, in which case theperturbation in the output field from the SLM may be represented asH₁(u,v) f(u₀,v₀) exp i φ₀(u₀,v₀).

As an example consider the perturbation to be a uniform change in phasemodulation over the area of a single square pixel of side p, from aninitial phase q₀ to a new phase q₁. This perturbation may be describedas a “flashing pixel”.

This is figuratively shown for a binary SLM in FIG. 3. For the sake ofclarity the majority of the pixels of the SLM are omitted. Blackindicates one sense of phase change and white the opposite sense. Thepixel at column 3 row 2 is the flashing pixel which is white for onestate of measurement and black for the other.

After the perturbation the phase modulation change applied to theselected pixel may revert to its original value and remain at that valuefor the rest of the test, for example while other pixels are flashed.For this example H₁(u,v) may be represented by equation (3):

$\begin{matrix}\begin{matrix}{{H_{1}\left( {u,v} \right)} = {{\exp\;{\mathbb{i}}\; q_{1}} - {\exp\;{\mathbb{i}}\; q_{0}}}} & {{\forall{{\left( {u,v} \right)\text{:}{{u - u_{0}}}} \leq \frac{p}{2}}},{{{v - v_{0}}} \leq \frac{p}{2}}} \\{= 0} & {elsewhere}\end{matrix} & (3)\end{matrix}$

Let the Fourier transform of H₁(u,v) be g(x,y)exp i ψ(x,y). Hence theFourier transform of H₁(u,v)f(u₀,v₀) exp i φ(u₀,v₀) is given by f(u₀,v₀)exp i φ(u₀,v₀) g(x,y) exp i ψ(x,y). So this perturbation field containsinformation about the phase and amplitude of the incident beam. For theexample given above, the perturbation field is given by equation (4):

$\begin{matrix}{{{{g\left( {x,y} \right)}\;{f\left( {u_{0},v_{0}} \right)}} = {2\;{\sin\left( \frac{q_{1} - q_{0}}{2} \right)}\; p^{2}\frac{\sin\left( {\pi\;{{px}/f}\;\lambda} \right)}{\pi\;{{px}/f}\;\lambda}\frac{\sin\left( {\pi\;{{py}/f}\;\lambda} \right)}{\pi\;{{py}/f}\;\lambda}{f\left( {u_{0},v_{0}} \right)}}}{{\exp\;{\mathbb{i}}\;{\psi\left( {x,y} \right)}\;\exp\;{\mathbb{i}}\;{\phi\left( {u_{0},v_{0}} \right)}} = {\exp\;{{\mathbb{i}}\left( {{\frac{2\;\pi}{f\;\lambda}\left\{ {{u_{0}x} + {v_{0}y}} \right\}} + \frac{\pi}{2} + q_{0} + \frac{q_{1} - q_{0}}{2} + {\phi\left( {u_{0},v_{0}} \right)}} \right)}}}} & (4)\end{matrix}$where f is the focal length of the Fourier lens. From the equation theamplitude of the perturbation field in the Fourier plane is proportionalto the amplitude of the incident field at the perturbation in thehologram. The phase of the perturbation field includes a constantcomponent equal to the phase of the incident field at the perturbationin the hologram, and also a linear component in x and y proportional tothe position of the perturbation in the hologram. This linear componentis due to the off-axis position of the perturbation and needs to betaken into account when interpreting phase measurements, as will bedescribed later. The flashing pixel can be considered as equivalent to asource radiating light. When the incident beam is normally incident, thecentre of the radiated beam travels parallel to the optical axis towardsthe Fourier lens, and is then incident at an angle towards thephotodetector. The angle of incidence is directly associated with thelinear term in equation (4). Note that this linear term disappears at(x,y)=(0,0), at the focal point of the Fourier lens. What happens whenthe incident beam is incident away from the normal direction isdiscussed later.

Returning to the general case, in the Fourier plane the field is theFourier transform of the output field from the SLM. Given that a Fouriertransform is a linear operation the field is the sum of the individualFourier transforms of the original and perturbation field from the SLMHence the total field in the Fourier plane, F(x,y) is that given by (5):F(x,y)=F ₀(x,y)exp iθ(x,y)+f(u ₀ ,v ₀)exp iφ(u ₀ ,v ₀)g(x,y)expiψ(x,y)  (5)

Light from the SLM is detected by a detector placed in the output fieldof the SLM, and typically disposed at a fixed location determinedempirically as being within the beam. The analysis given hereafterassumes the detector is a single photodiode that is small enough suchthat the field amplitude and phase in the Fourier plane may beconsidered uniform over the active area of the photodiode, or that thephase is varying slowly across it such that the phase of the differencesignal term (to be described) may be ascertained. In practice this meansthat the spatial period of the difference signal should be at leasttwice the width of the receiving element. Equations are derived for theresponse and data fitting methods are described to measure the amplitudevariation and phase variation of the field incident on the spatial lightmodulator. As mentioned previously, other receiving elements could beused. Examples are an optical fibre (single mode or multimode) coupledto a photodiode. In the case of a single mode fibre a mode strippershould also be used. Other example receivers are a larger photodiode oran array of photodiodes. For each case, knowing the physics of thereceiving process (which involves a coupling efficiency calculation forthe optical fibre case) analytical expressions may be used to derive thereceiver response, given the incident field as described in equation(5). As is demonstrated for the case of a small single photodiode, datafitting methods may be derived, based on said analytical expressions, tomeasure the amplitude variation and phase variation of the fieldincident on the spatial light modulator.

Assuming the field is detected directly by one or more photodiodes, theinduced photocurrent at position (x,y) is proportional to the localintensity. The intensity at the Fourier plane contains 3 terms.

The expression for the intensity, I(x,y), is given by equation (6):

$\begin{matrix}{{I\left( {x,y} \right)} \propto {{F_{0}^{2}\left( {x,y} \right)} + {2{F_{0}\left( {x,y} \right)}\;{g\left( {x,y} \right)}\;{f\left( {u_{0},v_{0}} \right)}\;\cos\left\{ {{\phi\left( {u_{0},v_{0}} \right)} + {\psi\left( {x,y} \right)} - {\theta\left( {x,y} \right)}} \right\}} + {{f^{2}\left( {u_{0},v_{0}} \right)}\;{g^{2}\left( {x,y} \right)}}}} & (6)\end{matrix}$

It is assumed that the beam incident on the SLM is spatially coherent,leading to constructive interference at the output field, such that theperturbation in the output intensity distribution is a small effect andsecond order effects in the perturbation may often be neglected.

The first term F₀ ²(x,y) is the original intensity, before theperturbation was applied. This term is conveniently described as areference signal.

The difference signal is calculated by subtracting the reference signalfrom the measured signal when the perturbation has been applied. Thisdifference signal contains the second and third terms in equation (6).In practice if the difference signal is weak compared to ambient noiselevels, it may be increased by flashing groups of contiguous pixels, atthe expense of a reduced resolution in the measurement process.

The second term 2F₀(x,y)g(x,y)f(u₀,v₀)cos {φ(u₀,v₀)+ψ(x,y)−θ(x,y)} is acoherent coupling term between the original field in the Fourier plane,and the field component created there by the perturbation at the SLM.This second term contains information about the phase and amplitude ofthe beam incident on the SLM. For the example hologram perturbation thesecond term also contains information about the flashing pixel.

The third term f²(u₀,v₀)g²(x,y) is the intensity that would appear atthe Fourier plane if the perturbation was acting on its own, with theoriginal field removed. This third term contains information about theamplitude of the beam incident on the SLM, but not the phase. The valuesthat are to be extracted from the system are f(u₀,v₀) and φ(u₀,v₀). Theother “unknowns” are g(x,y) and ψ(x,y)-θ(x,y).

For a general case where the incident beam phase varies in an unknownway, usually what is desired is the relative amplitude variation of thebeam, but not the absolute amplitude. The terms F₀(x,y)g(x,y) and g(x,y)are independent of the position of the perturbation at the SLM so can beconsidered as a multiplying constant. The amplitude f(u₀,v₀) may bemeasured in several ways.

A first method is to measure the intensity in a region of the Fourierplane where F₀(x,y) is very weak but g(x,y) is relatively stronger suchthat only the third term is significant. The region of the appliedperturbation at the SLM is narrower than the region occupied by theincident beam, hence at the Fourier transform plane the region occupiedby the Fourier transform of the perturbation is broader than the regionoccupied by the Fourier transform of the original field. Hence byvarying the position on the SLM where the perturbation is applied a setof measurements of (g(x,y)f(u₀,v₀))² may be built up at differentpositions (u₀,v₀). By taking the square root of these measurementsvalues for the relative field amplitude at these positions may bederived.

In general there may be sidelobes or spurious diffraction orders in thebeam F₀(x,y) making it difficult to find such a region. Often it may bethat g(x,y) is relatively weak and F₀(x,y) is stronger. Therefore itbecomes appropriate to use the second term to measure the amplitude.Indeed the F₀(x,y) term acts as a coherent ‘amplifier’ to boost thestrength of this second term. In such a region the third term may beneglected and the second term estimated by subtracting the receiveroutput before the perturbation was applied from the receiver output inthe presence of the perturbation.

A second method is to step through a sequence of distributions H₀(x,y)chosen to have the same values for F₀(x,y) and known changes in thevalues for θ(x,y). In an embodiment example H₀(x,y) is a binary phasepattern, in which F₀(x,y) is independent of the relative position of thepattern on the SLM, but θ(x,y) changes in a known way as the patternposition is changed on the SLM. In general let θ(x,y) be expressed asgiven by equation (7):θ(x,y)=θ₀+θ(m)  (7)where m is a variable that represents the known phase associated withthe pattern position, and θ₀ represents the sum of the background phasecomponent from the hologram that is not affected by the pattern positionand the phase from the incident field.

If this binary phase pattern, H₀(x,y), is periodic then it may beapplied to route the reflected beam to the photodiode or other receivingelement used to take the measurements. A multiphase periodic patterncould also be used for H₀(x,y). The multiphase method has the advantageof greater diffraction efficiency, but the binary phase method may beimplemented on a ferroelectric SLM which tends to be faster.

By applying a set of patterns, at different relative positions on theSLM a set of values may be built up for this second term (from 6).Substituting in equation (7) the term becomes that given in equation(8):2F ₀(x,y)g(x,y)f(u ₀ ,v ₀)cos {φ(u ₀ ,v ₀)+ψ(x,y)−θ₀(x,y)−θ(m)}  (8)which may be further expressed as given by equation (9). Note that for abinary phase SLM, the phase change at the pixel where we are measuringthe field, q₁−q₀ is +π or −π and therefore the term sin(½(q₁−q₀)) inequation (4) has the value +1 or −1.α(x,y)f(u ₀ ,v ₀)cos {β(x,y,u ₀ ,v ₀)} cos(θ(m))+α(x,y)f(u ₀ ,v ₀)sin{β(x,y,u ₀ ,v ₀)} sin(θ(m))  (9)where α(x,y)=2F ₀(x,y)g(x,y)  (10)β(x,y,u ₀ ,v ₀)=φ(u ₀ ,v ₀)+ψ(x,y)−θ₀(x,y).  (11)

Equation (9) may be considered as a linear equation in cos(θ(m)) andsin(θ(m)), with unknown coefficients c=α(x,y)cos β(x,y,u₀,v₀)f(u₀,v₀)and d=α(x,y)sin β(x,y,u₀,v₀)f(u₀,v₀) Any suitable data fitting methodmay be used to extract values for these coefficients c and d. A relativevalue for the amplitude may then be calculated form equation (12)(remembering that α(x,y) acts like a multiplying constant at any Fourieroutput position (x,y): successive measurements at different (u₀,v₀) willbe subject to the same value of multiplying constant).α(x,y)f(u ₀ ,v ₀)=√{square root over (c ² +d ²)}  (12)

Other methods could also be used to process the data. The third termfrom eqn (6) (quadratic in f(u₀,v₀)) could also be included in the datafitting.

In an environment with low noise, it may be suitable to use only twopattern positions, and hence two known values of θ(m), then using themethod of simultaneous equations to calculate c and d. The third termfrom (6) may also be included to improve accuracy, in which case wewould need three known values of θ(m). While the above example assumesan invariant F₀(x,y) and known changes in θ(x,y) hologram patterns couldbe selected that maintain an invariant θ(x,y) and change F₀(x,y) in aknown way, or that change both in a known way. Another method is tochange the amplitude and/or the phase of the perturbation hologram in aknown way.

This latter method of changing the phase in a known way is suitable fora multiphase level SLM. This is a third method to measure the amplitudevariation in the incident beam.

Let the SLM apply a uniform phase modulation of q₀ except at the flashedpixel where the phase applied is q₁. From equations (4) and (6), thecoherent coupling term becomes:

$\begin{matrix}{2{K\left( {x,y} \right)}\;{f\left( {u_{0},v_{0}} \right)}\;{\sin\left( \frac{q_{1} - q_{0}}{2} \right)}\;{\cos\left( {{\tau\left( {x,y,u_{0},v_{0}} \right)} + \frac{q_{1} - q_{0}}{2}} \right)}} & (13) \\{where} & \; \\{{K\left( {x,y} \right)} = {2{F_{0}\left( {x,y} \right)}\frac{p^{2}}{f\;\lambda}\frac{\sin\left( {\pi\;{{px}/f}\;\lambda} \right)}{\pi\;{{px}/f}\;\lambda}\frac{\sin\left( {\pi\;{{py}/f}\;\lambda} \right)}{\pi\;{{py}/f}\;\lambda}}} & (14) \\{{\tau\left( {x,y,u_{0},v_{0}} \right)} = {{\phi\left( {u_{0},v_{0}} \right)} + {\frac{2\;\pi}{f\;\lambda}\left( {{u_{0}x} + {v_{0}y}} \right)} + \frac{\pi}{2} + q_{0} - {\theta\left( {x,y} \right)}}} & (15)\end{matrix}$

Simplifying, we obtain a coherent coupling term given by:K(x,y)f(u ₀ ,v ₀){ cos(τ(x,y,u ₀ ,v ₀))sin(q ₁ −q ₀)−sin(τ(x,y,u ₀ ,v₀))(1−cos(q ₁ −q ₀))}  (16)

Clearly this term disappears as the perturbation in the local phase,q₁−q₀, tends to zero (or pi).

In a low noise environment we may measure the output with two sequentialdifferent known phase perturbations, compare with the output in theabsence of the phase perturbation and then use simultaneous equations tocalculate the values of K(x,y)f(u₀,v₀)cos(τ) and K(x,y)f(u₀,v₀)sin(τ).Squaring and adding and then taking the square root we obtainK(x,y)|f(u₀,v₀)|.

If we set the two phase perturbations to +pi/2 and −pi/2 the coherentcoupling term becomes:C(+)=K(x,y)f(u ₀ ,v ₀){ cos(τ)−sin(τ)}  (17)C(−)=K(x,y)f(u ₀ ,v ₀){−cos(τ)−sin(τ)}  (18)

Taking the sum of the square of C(+) and the square of C(−) and thentaking the square root we also obtain K(x,y)|f(u₀,v₀)|.

The term K(x,y) depends on the photodiode position and not the flashingpixel position. Hence a set of measurements of K(x,y)f(u,v) at differentpixel positions describes the relative variation in the incidentamplitude at the SLM.

Further we may obtain K(x,y)f(u₀,v₀)sin(τ) from C(+)+C(−) andK(x,y)f(u₀,v₀)cos(τ) from C(+)−C(−) and then divide to obtain tan(τ).How to use the value of τ to measure the incident phase φ(u₀,v₀) will bediscussed later.

This method of changing the phase at one pixel, while maintaining aconstant phase elsewhere, may also be used to characterise the phasemodulation of the SLM. Typically the phase vs. volts response of aliquid crystal SLM may be characterised by measuring output intensitychanges when the device is intermediate between crossed or parallelpolarisers oriented at 45 degrees to the plane of tilt of the liquidcrystal director, and the applied voltage is varied. However, suchintensity changes disappear when the SLM contains a QWP designed andfabricated for the wavelength at which the phase response is required.An alternative is to use Young's double slit to characterise the phase,observing the fringes created in the Fourier plane with a detector arrayand measuring how the fringe peaks shift as the voltage applied to thephase modulating pixels behind one of the slits is varied. Aligning theslits with a small pixel array can be time consuming and increase thecharacterisation time. Often there are awkward design tradeoffs betweenthe slit separation and the width of the slits. The difference signaldescribed in equation (13) may also be used to measure the phase. Thereference signal is F₀ ²(x,y). Hence dividing the difference signal bythe square root of the reference signal we obtain the normaliseddifference signal given in equation (19), where c is a constant ofproportionality.

$\begin{matrix}{{{cf}\left( {u_{0},v_{0}} \right)}\;\frac{p^{2}}{f\;\lambda}\frac{\sin\left( {\pi\;{{px}/f}\;\lambda} \right)}{\pi\;{{px}/f}\;\lambda}\frac{\sin\left( {\pi\;{{py}/f}\;\lambda} \right)}{\pi\;{{py}/f}\;\lambda}\;{\sin\left( \frac{q_{1} - q_{0}}{2} \right)}\;{\cos\left( {\tau + \frac{q_{1} - q_{0}}{2}} \right)}} & (19)\end{matrix}$

The output is measured with a 1-D or 2-D photodiode array. Keep all thepixels apart from the one or group to be flashed at a reference voltage:in the analysis these have an associated phase modulation q₀. Select oneor a group of pixels and apply a different voltage: in the analysisthese have an associated phase modulation q₁. The aim of the methoddescribed is to measure how the relative phase q₁−q₀ varies as afunction of the voltage applied to the flashing pixel. Vary the selectedpixel or group of pixels in order to maximise the strength of thenormalised difference signal. This selected pixel or group of pixelswill correspond to the peak of the incident beam, which is wheref(u₀,v₀) is maximised. The phase term τ naturally varies across thearray because of the linear phase term in (15), unless the flashingpixel happens to coincide with the optical axis of the lens. Hence themeasured output is a set of sinusoidal fringes inside a slowly varyingsinc envelope, with an amplitude proportional to sin((q₁−q₀)/2). Thefringe period may be conveniently optimised by adjusting the position ofthe beam incident on the SLM, with resect to the optical axis of theFourier lens. For example at a wavelength of 1.5 um, with a Fourier lensof focal length 15 cm and u₀=1 mm, the period at the photodiode arraywould be 232 um, covering 9 pixels of a 25 um pitch photodiode array.Note that the width of the sinc envelope is created by the width of theflashing pixel or group of pixels. As the width of this group isnarrower than the beam the envelope created will be broader than thesignal F₀(x,y). Hence dividing by the square root of the referencesignal increases the number of measurable fringes. For example with thesame wavelength and Fourier lens focal length, and a pixel group ofwidth 200 um, the width of the sinc main lobe would be 2325 um,containing 93 pixels, or 10 complete fringes.

Adjust the voltage applied to the flashing pixel or group of flashingpixels until the fringe amplitude is maximised. This identifies thevoltage creating a relative phase of π. In practice there may be afinite set of available voltages. Therefore data fitting may be used toidentify the actual voltage and fringe amplitude corresponding to arelative phase of π. Let this fringe amplitude be A. Therefore bydividing the normalised reference signal by A for a range of voltagesapplied to the flashing pixel, we obtain the value of sin((q₁−q₀)/2),from which we may calculate q₁−q₀ for the flashing pixel.

It is important in such measurements to use a stable input beam. Ifthere are slowly varying (temporal) changes in the input beam, it isnecessary to take fresh measurements of the reference signal in betweeneach measurement of the difference signal.

Returning to methods for measuring the incident amplitude, the generalprinciple is to be able to take a sequence of measurements with someknown parameters changing in a known way and use data fitting to extractthe values of the unknown parameters. These unknown parameters may becombined in a way that extracts the incident field amplitude withoutrequiring knowledge of the incident field phase.

Another measurement that may be required is to check that the pixels arefunctioning correctly. Take a measurement of the output intensity withthe original hologram applied: this could be the coupling efficiencyinto a fibre, or the output from a photodiode, or the outputs from aphotodiode array. Now flash a pixel, one at a time, preferably with aphase perturbation q₁−q₀ close to π so as to maximise the effect.Calculate the relative amplitude as described above. If there is nosignificant amplitude then either the pixel is outside the area of theincident beam or the pixel drive circuit is not functioning. If theamplitude varies in a smooth manner like the expected profile then thepixels are working correctly. If there is a discontinuous jump in theresponse to a higher value then the pixel may be shorted to anotherpixel. If there is a discontinuous jump to a lower value then the pixelis not being driven correctly.

In some systems what is required is to check or measure the position ofthe field with respect to the SLM pixels. The peak measured amplitudewill occur when the pixel being flashed coincides with the peak of theincident beam.

In many cases it may also be required to measure the phase variation inthe incident field. The methods described hereafter explain how tomeasure the variation in the u direction in the interests of clarity.However the method is easily extended to measurements in both u and vdirections.

In the first case we consider the example of a binary phase SLM wheresuccessive measurements are obtained at a single flashed pixel byshifting the position of the pattern H₀(x,y) on the SLM.

Given the data fitted values of the coefficients c and d the phase termβ(x,y,u₀,v₀) may be calculated using equation (20).

$\begin{matrix}{{{\cos\left( {\beta\left( {x,y,u_{0},v_{0}} \right)} \right)} = \frac{c}{{\alpha\left( {x,y} \right)}\;{f\left( {u_{0},v_{0}} \right)}}}{{\sin\left( {\beta\left( {x,y,u_{0},v_{0}} \right)} \right)} = \frac{d}{{\alpha\left( {x,y} \right)}\;{f\left( {u_{0},v_{0}} \right)}}}} & (20)\end{matrix}$

In the second case we consider the example of a multiphase SLM where theoriginal or reference pattern is uniform phase modulation. The values ofK(x,y)f(u₀,v₀)cos(τ) and K(x,y)f(u₀,v₀)sin(τ) may be calculated asdescribed earlier and solved jointly to obtain τ(x,y,u₀,v₀).

In what follows we discuss how to interpret the value of β, but thediscussion and methods apply equally to the value of τ.

Let the initial position (u₀,v₀) be a reference point such that thephase at any arbitrary point (u,v) is measured with respect to the phaseat (u₀,v₀). Using (4) there is obtained a general expression forβ(x,y,u₀,v₀) as shown in equation (21):

$\begin{matrix}{{\beta\left( {x,y,u_{0},v_{0}} \right)} = {\frac{2\;\pi\; u_{0}x}{f\;\lambda} + {\phi\left( {u_{0},v_{0}} \right)} + \left\{ {\frac{2\;\pi\; v_{0}y}{f\;\lambda} + \frac{\pi}{2} + \frac{q_{0} + q_{1}}{2} - {\theta_{0}\left( {x,y} \right)}} \right\}}} & (21)\end{matrix}$

Now consider the effect of changing the position of the hologramperturbation to (u,v₀) but maintaining the position of the outputreceiver at (x,y). The value of β becomes that shown in equation (22):

$\begin{matrix}{{\beta\left( {x,y,u,v_{0}} \right)} = {\frac{2\;\pi\;{ux}}{f\;\lambda} + {\phi\left( {u,v_{0}} \right)} + \left\{ {\frac{2\;\pi\; v_{0}y}{f\;\lambda} + \frac{\pi}{2} + \frac{q_{0} + q_{1}}{2} - {\theta_{0}\left( {x,y} \right)}} \right\}}} & (22)\end{matrix}$

The difference in the values of β contains two terms, the first (incurly brackets) is the required phase difference being measured, whilethe second is a linear phase term, as shown in equation (23):

$\begin{matrix}{{{\beta\left( {x,y,u,v_{0}} \right)} - {\beta\left( {x,y,u_{0},v_{0}} \right)}} = {\left\{ {{\phi\left( {u,v_{0}} \right)} - {\phi\left( {u_{0},v_{0}} \right)}} \right\} + \frac{2\;\pi\;\left( {u - u_{0}} \right)\; x}{f\;\lambda}}} & (23)\end{matrix}$

Therefore to extract the required phase difference a method is needed toestimate or measure the linear phase term. Given knowledge of the pixelpitch on the SLM it is straightforward to calculate (u−u₀). Similarlyknowledge of the wavelength, λ, is established. The focal length can bemeasured. Therefore it is possible to know to a high degree of accuracythe value of 2π(u−u₀)/fλ. What may be difficult is to obtain an accuratevalue of x, the distance in the x direction from the point where thelens optical axis intersects the Fourier plane to the receiving element.In some situations it may be possible to place the receiving element atthe focal point in which case the linear term is zero.

One method to overcome this difficulty is to pre-calibrate the systemwith a reference beam. This could be a well-collimated beam, or a beamof known phase variation (using for example a pinhole to generate thereference beam), or a beam such that the phase of any arbitrary beamneeds to be measured with respect to the reference beam. Let thesubscript R represent the corresponding measurements for the referencebeam. Hence equation (24) is obtained:

$\begin{matrix}{{{\beta_{R}\left( {x,y,u,v_{0}} \right)} - {\beta_{R}\left( {x,y,u_{0},v_{0}} \right)}} = {\left\{ {{\phi_{R}\left( {u,v_{0}} \right)} - {\phi_{R}\left( {u_{0},v_{0}} \right)}} \right\} + \frac{2\;{\pi\left( {u - u_{0}} \right)}x}{f\;\lambda}}} & (24)\end{matrix}$

Therefore if the β difference is measured for the reference beam and thephase variation across the reference beam is known, the value of thelinear phase term may be calculated. Furthermore, by repeating thisprocess for several different positions u on the SLM a data fittingmethod may be used to calculate the value of the unknown parameter x.

Without knowledge of the phase variation of the reference beam but witha need to use it as a baseline, it is necessary to measure the βdifference for the beam under test, and subtract from it the βdifference for the reference beam, at the same positions u and u₀.Alternatively interpolation may be used to predict the β difference forthe reference beam if it has not been measured at precisely the requiredpoints. An expression may be obtained for the phase variation withrespect to the baseline variation, as shown in equation (25):

$\begin{matrix}{{{\beta\left( {x,y,u,v_{\; 0}} \right)} - {\beta\left( {x,y,u_{\; 0},v_{\; 0}} \right)} - \left\{ {{\beta_{R}\left( {x,y,u,v_{0}} \right)} - {\beta_{R}\left( {x,y,u_{0},v_{0}} \right)}} \right\}} = {{\phi\left( {u,v_{0}} \right)} - {\phi\left( {u_{0},v_{0}} \right)} - \left\{ {{\phi_{R}\left( {u,v_{0}} \right)} - {\phi_{R}\left( {u_{0},v_{0}} \right)}} \right\}}} & (25)\end{matrix}$

Another suitable reference beam is a Gaussian beam from a IR HeNe laseror another beam with a well-defined main peak at the Fourier plane. Onemethod is to arrange that the beam is normally incident on the SLM, andthat the SLM is perpendicular to the optical axis of the Fourier lens.Let uniform phase modulation be applied by the SLM such that it actslike a plane minor. In this case the peak of the reflected beam willreach the Fourier plane at the focal point, thus determining theposition where x=0. What is required is to measure the x position of thephotodetector or other receiving element Beam-steering holograms ofknown beam deflection d in the x direction may be applied by the SLM andthe incident power measured at the receiving element The value of d atwhich this power is maximised is equal to the x displacement of thereceiving element from the optical axis, therefore determining theunknown parameter x.

The peak of the characterised beam, (u_(PK),v_(PK)) may be identified bydata fitting or centroid methods on the amplitude distribution, wherethis distribution may be measured as described earlier. Adapting theexpression for the β difference (eqn 23) the relative phase variationmeasured with respect to this peak becomes that shown in equation (26)where it is assumed as described earlier, that u−u_(PK), f and λ areknown and x has been measured by one or more of the methods described:

$\begin{matrix}{{{\beta\left( {x,y,u,v_{0}} \right)} - {\beta\left( {x,y,u_{PK},v_{0}} \right)}} = {\left\{ {{\phi\left( {u,v_{0}} \right)} - {\phi\left( {u_{PK},v_{0}} \right)}} \right\} + \frac{2\;\pi\;\left( {u - u_{\;{PK}}} \right)x}{f\;\lambda}}} & (26)\end{matrix}$

In subsequent wavefront sensing tests, the linear phase term in (23)(and (26)) may be calculated using the measured value of x andsubtracted from the β difference to obtain the relative phase variationacross the beam under characterisation, φ(u,v)−φ(u_(PK),v). Generalisingto the 2D case we may also determine the y position of the receivingelement, and calculate the relative phase variation with respect to thecentre of the beam, φ(u,v)−φ(u_(PK),v_(PK)). The partial derivativeswith respect to u and v of the phase at u=u_(PK) and v=v_(PK) may becalculated from the relative phase variation and used to determine theabsolute angle of incidence of the beam upon the SLM.

In a more general case a beam may be incident at an angle, γ, withrespect to the optical axis. This angle of incidence creates a linearphase slope, 2π sin(γ)/λ at the flashing pixel. Prom Fourier theory (andalso from geometric optics) this phase slope creates an offset at theFourier plane. Hence the field generated by the flashing pixel isdisplaced along the x-axis by x₀=f tan(γ). The angle of incidence alsocreates a linear phase slope across the original hologram pattern,H₀(u,v). For a normally incident beam let the FT of the field reflectedfrom the original hologram, that is the reference field, beF₀(x,y).expiφ(x,y). The effect at the Fourier plane of changing theangle of incidence to γ is to change the reference field toF₀(x−x₀,y).expiθ(x−x₀,y), representing a displacement of x₀ along the xaxis. The difference signal is created by the coherent overlap of theflashing pixel field and the reference field. Therefore the differencesignal is also displaced along the x axis by an amount x₀. Hence the βdifference term in equation (26) becomes that given in equation (27):

$\begin{matrix}{{{\beta\left( {x,y,u,v_{0}} \right)} - {\beta\left( {x,y,u_{PK},v_{0}} \right)}} = {\left\{ {{\phi\left( {u,v_{0}} \right)} - {\phi\left( {u_{PK},v_{0}} \right)}} \right\} + \frac{2\;\pi\;\left( {u - u_{PK}} \right)\left( {x - x_{0}} \right)}{f\;\lambda}}} & (27)\end{matrix}$

In this expression, the term in the curly brackets presents the relativephase variation of the incident beam, for the case of normal incidence,while the linear term is created by the angle of incidence. Hence thecurly bracket term represents the beam defocus, decoupled from angle ofincidence effects.

In the absence of the flashing pixel perturbation, known beamingpatterns may be applied by the SLM to measure the x displacement of thereceiving element from x₀, in order to calculate x−x₀. A linear phaseterm 2π(u−u_(PK))(x−x₀)/fλ may be calculated and subtracted from the βdifference in equation (27) to obtain the curly bracket term, and hencethe beam defocus.

Given the value of x, as measured using one of the methods describedearlier, the angle of incidence may be extracted from the phase result.This may be achieved by subtracting 2π(u−u_(PK))x/fλ from the βdifference given in (27) and calculating the is partial derivatives ofthe result with respect to u and v at the beam centre.

It is useful to extend this analysis to consider the effect of beamdeflection by a beam-steering hologram. In this example the originalhologram pattern, H₀(u,v) is a routing hologram that creates a set ofdiscrete diffraction orders positioned along the x-axis as described inequation (28):

$\begin{matrix}{x_{n} = {x_{0} + \frac{{nf}\;\lambda}{\Lambda}}} & (28)\end{matrix}$where Λ is the beam steering period and x₀ may be non-zero, due to anon-zero angle of incidence at the SLM. Let F₀(x,y)exp i θ₀(x,y)represent the field at the Fourier plane when the SLM is applying auniform phase modulation. Also let the phase and amplitude of thediffraction order used to route to the receiving element be θ_(R) andc_(R), respectively. Hence the routing hologram creates a diffractionorder described by c_(R) F₀(x−x_(n),y)exp i {θ₀(x−x_(n),y)+θ_(R)}. Inthe first method of using a routing hologram, variations in thedifference field were created by shifting the routing hologram acrossthe SLM in order to change the phase of the diffraction order, and thuschange the phase difference with respect to the flashing pixel field. Inthis second method let the variations in the difference field be createdby changing the phase difference applied by the flashing pixel. Hencethe relevant phase term to consider is τ, although the analysis canequally be applied to β. The relevant expression for τ is given inequation (29):

$\begin{matrix}{{\tau\left( {x,y,u,v} \right)} = {{\frac{2\pi}{f\;\lambda}\left\{ {{u\left( {x - x_{0}} \right)} + {vy}} \right\}} + \frac{\pi}{2} + {q_{0}\left( {u,v} \right)} + {\phi\left( {u,v} \right)} - {\theta_{0}\left( {{x - x_{n}},y} \right)} - \theta_{R}}} & (29)\end{matrix}$

Note that the original hologram phase, q₀, now depends on the pixelposition, since the original hologram pattern is a routing hologram,rather than a uniform phase distribution.

The relative variation in τ with respect to the beam centre at(u_(PK),v_(PK)) becomes that given in (30):

$\begin{matrix}{{{\tau\left( {x,y,u,v} \right)} - {\tau\left( {x,y,u_{PK},v_{PK}} \right)}} = {{\frac{2\pi}{f\;\lambda}\left\{ {{\left( {u - u_{PK}} \right)\left( {x - x_{0}} \right)} + {\left( {v - v_{PK}} \right)y}} \right\}} + {\phi\left( {u,v} \right)} - {\phi\left( {u_{PK},v_{PK}} \right)} + {q_{0}\left( {u,v} \right)} - {q_{0}\left( {u_{PK},v_{PK}} \right)}}} & (30)\end{matrix}$

The phase component q₀(u,v)−q₀(u_(PK),v_(PK)) is a known quantity andhence may be subtracted to obtain the expression given in equation (31):

$\begin{matrix}{{{\tau\left( {x,y,u,v} \right)} - {\tau\left( {x,y,u_{PK},v_{PK}} \right)} - \left\{ {{q_{0}\left( {u,v} \right)} - {q_{0}\left( {u_{PK},v_{PK}} \right)}} \right\}} = {{\frac{2\pi}{f\;\lambda}\left\{ {{\left( {u - u_{PK}} \right)\left( {x - x_{0}} \right)} + {\left( {v - v_{PK}} \right)y}} \right\}} + {\phi\left( {u,v} \right)} - {\phi\left( {u_{PK},v_{PK}} \right)}}} & (31)\end{matrix}$

The result on the right hand side of this equation is a 2D extension ofthat in (27), but is otherwise isomorphic. Hence the beam deflectioncreated by the hologram does not corrupt the phase measurement.

The data processing methods analysed so far have assumed detection witha single photodiode or an array of photodiodes. It is also possible tomeasure the amplitude and phase at the SLM by detecting the output powerfrom a single-mode fibre or waveguide. One advantage of such a method isthat there is no need for a beamsplitter, which necessarily introducesan insertion loss. Hence the waveguide method may be used for anassembled system: the actual waveguide used could be one of the normaldevice ports, or a separate port used specifically for sensing. A secondadvantage is that it could be used for remote sensing, using fibresrouted to a set of different sensors, with a common head-end fordetecting and processing the signals returned from the sensor.

The power, P, coupled into the fundamental mode of a single-modewaveguide may be measured with a photodiode, and is given by equation(32), in which η₂(x,y) is the field distribution (assumed real) of thefundamental mode, and η₁(x,y) is the complex field incident on saidwaveguide:

$\begin{matrix}{P = \frac{{{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{\eta_{1}^{*}\left( {x,y} \right)}{\eta_{2}\left( {x,y} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}}^{2}}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{\eta_{2}\left( {x,y} \right)}}^{2}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}} & (32)\end{matrix}$

Assume that the field routed towards said waveguide by the originalhologram is well aligned with that waveguide and has a beam profilesuitable for good coupling efficiency into the waveguide. Hence we mayapproximate:

$\begin{matrix}{{F_{0}\left( {x,y} \right)} \approx {{A\mspace{11mu}\exp} - \left\{ \frac{\left( {x - x_{F}} \right)^{2} + y^{2}}{\omega^{2}} \right\}}} & (33) \\{{\theta_{0}\left( {x,y} \right)} \approx \theta_{0}} & (34)\end{matrix}$where ω is the spot size of the fundamental mode, and (x_(F),0) are theco-ordinates of the centre of the mode. The method may also be appliedfor other cases of beam shape and alignment, although the analysis ismore complex.

When the original hologram pattern is applied the coupled power is givenby P₀=Aπα²/2, which forms the reference signal.

The flashing pixel field with a non-zero angle of incidence may besubstituted into (32) to calculate the difference signal in the coupledpower that is created by the flashing pixel. Using the approximationthat the sinc terms in the overlap integral of (32) are slowly varyingand may be taken outside the integral, the result for a flashing pixelapplied at position (u,v) is given by the expression in equation (35):

$\begin{matrix}{{{K^{\prime}\left( {x,y} \right)}\exp} - {\left\{ \frac{u^{2} + v^{2}}{\omega_{SLM}^{2}} \right\}{f\left( {u,v} \right)}{\sin\left( \frac{q_{1} - q_{0}}{2} \right)}{\cos\left( {{\chi\left( {u,v} \right)} + \frac{q_{1} - q_{0}}{2}} \right)}}} & (35)\end{matrix}$where K′(x,y) varies slowly with (x,y), ω_(SLM) is the beam spot size atthe SLM, and χ(u,v) is given in equation (36):

$\begin{matrix}{{\chi\left( {u,v} \right)} = {\frac{2\pi\;{u\left( {x_{F} - x_{0}} \right)}}{f\;\lambda} + {\phi\left( {u,v} \right)} + {q_{0}\left( {u,v} \right)} - \theta_{0}}} & (36)\end{matrix}$and x₀ is the output position, at the Fourier plane, of the specularlyreflected beam when the SLM is acting as a uniform phase object normalto the optical axis. By varying (q₁−q₀/2) as described earlier, valuesmay be extracted for χ(u,v) and also for the amplitude term given inequation (37):

$\begin{matrix}{{{K^{\prime}\left( {x,y} \right)}\exp} - {\left\{ \frac{u^{2} + v^{2}}{\omega_{SLM}^{2}} \right\}{f\left( {u,v} \right)}}} & (37)\end{matrix}$

The value of x_(F)−x₀ is related to the beam steering period at thehologram, Λ, as shown in equation (28). Assuming the first diffractionorder is used to route to the waveguide, the expression for χ(u,v) maybe written as in equation (38):

$\begin{matrix}{{\chi\left( {u,v} \right)} = {\frac{2\pi\; u}{\Lambda} + {\phi\left( {u,v} \right)} + {q_{0}\left( {u,v} \right)} - \theta_{0}}} & (38)\end{matrix}$

Hence the relative variation in χ(u,v) may be calculated, as shown inequation (39):

$\begin{matrix}{{{\chi\left( {u,v} \right)} - {\chi\left( {u_{0},v} \right)}} = {\left\{ {{\phi\left( {u,v} \right)} - {\phi\left( {u_{0},v} \right)}} \right\} + {q_{0}\left( {u,v} \right)} - {q_{0}\left( {u_{0},v} \right)} + \frac{2{\pi\left( {u - u_{0}} \right)}}{\Lambda}}} & (39)\end{matrix}$

The difference in q₀ and the linear term are known quantities andtherefore may be subtracted to calculate the relative phase variationacross the beam incident on the SLM.

The expression for the amplitude in (37) is interesting. The middle termin this expression is a result of the dependence of the waveguidecoupling efficiency on the angle of incidence of the incident beam, andis absent or much weaker for a photodiode. To extract the requiredamplitude variation, f(u,v), requires an additional calibration step,which is to determine the absolute values of u and v for the SLM pixels.

A method to achieve this is illustrated in FIG. 7, in which abeamsplitter 701 is inserted between the waveguide 702 and Fourier lens703. In this example the input beam 704 is incident on the SLM 705 viathe Fourier lens 703 and is shown at a non-zero angle of incidence. Asuitable input beam would be that coming from the sensing waveguide 702or another waveguide, preferably parallel to the sensing waveguide. Thereflected beam 706 is steered by the beam-steering hologram displayed onthe SLM to the waveguide 702. A fraction of the beam energy 707 is splitoff by the beamsplitter 701 towards the photodiode 708. Wavefrontsensing at the photodiode may be used to measure K(x,y)f(u,v) as afunction of u and v. Data fitting to the result may be used to extractthe pixel addresses of the centre of the beam as it reaches the SLM, andalso the spot size at the SLM, ω_(SLM). Wavefront sensing at thewaveguide may be used to measure the amplitude term given in (37). Theratio of the amplitude measurements at the photodiode and waveguideyields the value of exp−{(u²+v²)/ω_(SLM) ²}. Data fitting to the resultmay be used to extract the pixel address of the position (u,v)=(0,0),that is where the optical axis of the lens intersects the SLM. Hence, asrequired for subsequent sensing tests, the absolute values of u and vmay be calculated.

Further, this calibration process indicates whether the centre of thebeam at the SLM coincides with the optical axis, providing usefulinformation that may be fed back to alignment equipment that adjusts thetilt of the waveguide or waveguides until the centre of the beam doescoincide with the optical axis, at which point the input beam isparallel to that axis.

Having described how the wavefront sensing may be implemented we nowdiscuss how it may be applied to solve alignment problems.

In the first example an SLM or its pixel assignment is required to bealigned with a set of one or more beams. Some embodiments are shown inFIGS. 4, 5 and 6.

FIG. 4 shows an SLM 100 with two incident beams 101 and 102. Beam 101 isincident with its centre or another known feature of the beam atposition 103 on the SLM and beam 102 is incident with its centre oranother known feature of the beam at position 104 on the SLM. Abeamsplitter 105 is used to deflect a fraction of the reflected beams106 and 107 through a Fourier lens 112 to a photodiode, to photodiodesor other receiving element 108 connected to signal processing means 109.In one embodiment the beamsplitter, 105, is a cube beamsplitter of thetype that does not cause a transverse displacement of the beam. However,if the subsequent optical processing (after wavefront sensing) requiresthe beamsplitter to remain in position, it does not matter if itintroduces such a displacement.

If the two beams 101 and 102 are from different sources, or with asignificant relative delay so as to make them incoherent with respect toeach other, the field generated by the flashing pixels in the area ofthe SLM for which the incident amplitude of beam 101 is significant willbe coherent with the reflected beam 106, but incoherent with thereflected beam 107. As a result, a difference signal will be created bythe overlap of the flashing pixel field and the reflected beam 106, butnot by the overlap of the flashing pixel field and the reflected beam107. Hence the measured difference signal from this flashing pixel willnot be corrupted by the presence of the second beam, thereby allowingaccurate wavefront sensing of the first beam. However, if the two beamsto be sensed are from the same source or coherent then it may benecessary to switch one off at a time to avoid the generation ofadditional unwanted difference signals.

In a first alignment application what is required is to adjust thetransverse position of the SLM with respect to this set of one or morebeams. Wavefront sensing may be applied to measure the amplitudedistribution of the two beams as a function of the pixel address of theflashed pixels.

A first class of this application is suitable for Gaussian incidentbeams, or other beams with an even symmetric amplitude distribution. Forthis class of beams, data fitting or centroiding methods may be appliedto calculate the pixel address of the centres 103 and 104 of said beams.The results may be fed back from the signal processor 109 to a controlsystem 110 that compares these pixel addresses with the desiredaddresses for the centres of said beams, calculates how much the SLMneeds to be moved so that the centres of the beams are at the desiredaddresses, and adjusts the position of the SLM by that amount using anelectronically controlled positioning system 111. Alternatively, or aswell, the SLM position may be adjusted and the beam centre positionscompared with the desired positions until the beam centre positionsreach the desired positions.

A second class of this application is suitable for other incident beams,without even symmetry, but with known features in the beam (such as anedge created by an aperture). These features may be identified in themeasured amplitude distribution and data fitting or feature extractionused to calculate the location of these features as a pixel address.Again the results may be fed back from the signal processor 109 to acontrol system 110 that compares these pixel addresses with the desiredaddresses for these features, calculates how much the SLM needs to bemoved so that the features arrive at the desired pixel addresses, andadjusts the position of the SLM by that amount using an electronicallyor manually controlled positioning system 111. Alternatively, or aswell, the position may be adjusted and the feature positions 103 and 104compared with the desired positions until the feature positions reachthe desired positions.

In a second alignment application what is required is to assign orre-assign a block of pixels to the incident beams. Pixel blocks 113 and114 may be assigned originally to each beam. Each block of pixels is tobe used to perform optical processing on a particular beam. For thefirst class of beams, the calculated pixel addresses of the centres ofeach beam may be used to assign or re-assign the pixel addresses of theblock of pixels that is intended to perform optical processing on thatbeam. For the second class of beams, the calculated pixel addresses ofthe known features in the beam may be used to assign or re-assign thepixel addresses of the block of pixels that is intended to performoptical processing on the beam.

In a third alignment application what is required is to adjust thelongitudinal position of the SLM with resect to the beams.

In a first method the phase distribution (defocus) in one or moreincident beams may be measured using the wavefront sensing methodsdescribed, and the longitudinal position adjusted using a positioningsystem until said phase distribution reaches a target or optimisedvalue.

In a second method the measured phase distribution (defocus) may be usedto calculate how much the SLM should be moved longitudinally, and theposition of the SLM 100 adjusted using a manually or electronicallycontrolled positioning system until the target or an optimised phasedistribution is reached.

In a third method the transverse position at which two beams arrive onthe SLM may be extracted from a measurement of the amplitudedistribution, based on the beam centre or some known feature of thebeam, as described earlier. The longitudinal position of the SLM may beadjusted and the incident beams re-measured until their relativeposition reaches a target or optimised value.

In a fourth method the change in beam transverse position withlongitudinal position of the SLM may be measured and used to calculatehow much the SLM should be moved longitudinally, and the position of theSLM 100 adjusted using a manually or electronically controlledpositioning system until the target or an optimised relative position isreached.

In a fourth alignment application what is required is to adjust the tiltof an incoming beam. The angle of incidence on the SLM may be measuredas described earlier and the beam tilt adjusted until the required valueis reached. Alternatively or as well, the error in the angle ofincidence may be used to calculate the tilt adjustment required toachieve the target value.

FIGS. 5 and 6 show alternative embodiments for taking the aforementionedor other wavefront sensing measurements.

In FIG. 5 there is already a lens 200 in the path of the incident beams201 and 202 before said beams reach the SLM 203. The lens may be at theFourier plane of the SLM but does not have to be, as explained earlier.The beamsplitter may be positioned between the lens 200 and the SLM 203,as in the previous example. Alternatively the beamsplitter 204 may beplaced as in FIG. 5, such that the beams 205 and 206 reflected from theSLM 203 are deflected towards the receiving element 207 after passingthrough the lens 200. This method has the advantage that the relativealignment of the lens 200 and SLM 203 is not affected by displacementsand optical path differences introduced by the beamsplitter 204.

In a third arrangement FIG. 6) the pixel blocks 300 and 301 assigned tothe incident beams 302 and 303 on the SLM 304 are applying routingholograms, as described earlier, such that the reflected beams 305 and306 may be deflected towards the receiving element 307 without the needfor a beamsplitter. This receiving element 307 may be in the same planeor a different plane to the incident beam 308. The position of theFourier lens 308 shown in the Figure is such that both incident andreflected beams pass through the lens. However it could be positioned sothat only the beam steered towards the receiving element 307 passesthrough the lens.

Having described how alignment problems may be solved by using wavefrontsensing we now describe how to use wavefront sensing to monitor beamsentering an optical system that uses an SLM to perform opticalprocessing, and also how to control the assignment of pixel blocks tothe beams that are being processed.

In a first application (FIG. 8) it is assumed that there are one or morebeams 801, 802 entering the optical system from an optical fibre 803,each containing an ensemble of one or more signals at differentwavelengths. A diffraction grating or prism 804 close to the focal planeof a routing lens 805 is used to disperse the channels onto an SLM 806at the other focal plane of the lens. Two channels 807, 808 are shown onthe figure, Each wavelength signal has associated with it an individualpixel block 809, 810 to which various holograms are applied to route,equalise and block the beam. Hence there is a contiguous row of suchpixel blocks 811 across the SLM 806.

The different wavelength signals entering the system are incoherent withrespect to each other. Therefore the field generated by a flashing pixelin an SLM pixel block 809 receiving a first wavelength channel 807 willbe incoherent with the field reflected from an SLM pixel block 810receiving a second wavelength channel 808. Hence the overlap between theflashing pixel field and the field at the second wavelength will notgenerate a difference signal at the receiving element 812. Therefore thepresence of this second wavelength will not corrupt wavefront sensing ofthe first wavelength. Similarly, the beam at the second wavelength maybe sensed by flashing pixels in its own pixel block, withoutinterference from the first wavelength.

In a first method the flashing pixels are used to sense the beams thatare subsequently routed out of the optical system. In this example thereceiving element 812 will be the output optical fibre to which thechannels are routed. A monitor tap coupler in this optical fibre 813 maybe used to extract a fraction of the output signal and route this to aphotodiode 814 connected to signal processing circuitry 815 whichextracts the superposition of the difference signals for all thewavelength signals entering that output fibre.

In a second method (FIG. 9) the flashing pixels are used to sense a copyof the beams that are subsequently routed out of the optical system.Compared to the first method, this has the advantage that differencesignals are not added to the data carried by the wavelength channels.

A free space optical tap 901 is used to create a weak copy 902 of theinput beams at an angle to the remaining energy in the input beams 903,hereinafter referred to as the main input beams. The weak copy beams 902propagate to a separate row of pixel blocks 903, while the main inputbeams propagate to the row of pixel blocks 904 used for routing, channelblocking and channel equalisation. The copy of the input beams 902 maybe routed or specularly reflected, as desired, to a receiving element905, where overlap with fields generated by flashing pixels will createa superposition of difference signals.

In a third method (FIG. 10) a monitor tap 1007 in the output fibre 1001,possibly alter an optical amplifier 1002, is used to extract a weak copyof the routed beams after they have been routed through the opticalsystem. This weak copy may be injected back into that optical system tocreate a weak beam 1003 at an angle to the main input beam 1004, so thatthe copy arrives at a separate row of pixel blocks 1005, where flashingpixels are used to sense the beams. These beams may be routed orspecularly reflected, as desired, to a receiving element 1006.

For all three methods, control circuitry 817 is used to control andselect the flashing pixels so as to create flashing pixel fields withknown and desirable time dependence. This time dependence could includesequential polling of the different wavelength channels routed to thatoutput fibre, or continuous signals containing orthogonal codes, orcontinuous signals containing frequency tones at known but disjointfrequencies. The difference signals may be processed in the signalprocessing circuitry 815 to identify which wavelength channels arepresent. For example if sequential polling has been used, the presenceof difference signals during the time period for which the SLM isexpected to respond to the flash applied to the pixel block assigned tothat channel indicates the presence of the channel, while the absence ofsuch a signal indicates that the channel is unoccupied. This methodenables periodic checks on the channel occupancy but has the drawbackthat there is a delay of one cycle time (through all polled channels)before it is possible to know that a channel has become unoccupied.Further, two flashes are required to measure each channel.

Frequency tones in the difference signal may be detected with a set offilters. The presence of the tone indicates the presence of the channel,while the absence of such a tone indicates that the channel isunoccupied. The advantage of the frequency tone method is that themonitor signal for a particular channel is then continuous, reducing thetime to detect that the channel has become unoccupied.

As well as indicating occupancy, the strength of the difference signalcan indicate the relative power level in the channel for use in controlalgorithms to adjust the channel equalisation. Small changes in theposition of the flashing pixels may be applied to confirm the positionsof the centres of the beams as they reach the SLM, and the assignment ofpixel blocks to each channel adjusted accordingly.

The strength of the difference signal may be increased by flashing morethan one pixel in the same column. Adjacent columns may also be flashedif required to further increase the difference signal.

FIG. 11 shows a set of beams 1103 to 1107 incident on a 2D array ofpixels. Each beam carries the signal in a wavelength channel. 1101 is anindividual pixel and 1102 is a block of pixels assigned to a particularbeam. In the figure, columns of pixels 1108 to 1112 in each assignedblock are used to sense each individual beam. Further, the positions ofthe flashing pixels across the row of pixel blocks may be considered asa comb. The relative position of the comb compared to the incident beams1103 to 1107 may be adjusted slowly back and forth to detect changes inthe relative wavelength spacing between channels, and thus indicate if achannel has drifted, as has the channel creating beam 1105.

While the aforementioned monitoring applications have used the measuredamplitude distributions to extract information about the signals passingthrough the system, the wavefront sensing technique described herein mayalso detect phase distributions, as described earlier, and thedistribution processed to calculate angle of incidence and defocus. Inan adaptive optical system this information may be used to control thealignment of the optical beams passing through the system.

Some free space taps (FIG. 12) will create two copies 1201, 1202 of theinput beams. Preferably these should be at equal angles but on oppositesides of the main input beams 1203. The positions 1204, 1205 where thesecopies of the input beams arrive at the SLM 1206 may be measured asdiscussed under alignment techniques. Interpolation may be applied tocalculate the position 1207 where the main input beams 1203 are incidenton the SLM, and the assignment of pixel blocks 1208 to these main inputbeams adjusted without creating difference signals on top of the data.These two copies 1201, 1202 of the input beam may have the same ordifferent distributions of flashing pixels applied. Further the phaseperturbation applied at the flashing pixels may be the same ordifferent. The two reflected and sensed copy beams may pass back throughthe free-space tap 901 to separate receiving elements 1209 and 1210.

It may be advantageous to use more than one receiving element forwavefront sensing or monitoring applications. A first example is toallocate the wavelength channels to a set of groups, with each group ofchannels monitored at a particular receiving element.

Preferably the receiving elements are positioned and the copy beamsangled such that the field generated by the flashing pixel(s) does notcreate significant difference signals on top of the data in the mainoutput beams, if that is undesirable.

A second example is to monitor two channels at the same nominalwavelength in a system where they are incident on the same pixel block.Generally these two channels would be incident at different angles, sowhen specularly reflected or deflected with the same routing hologramwould be incident on different receiving elements. A first receivingelement should be positioned to receive a significant fraction of thebeam from the first channel but an insignificant fraction of the beamfrom the second channel. Hence the difference signal from this firstreceiving element is not influenced by the power in the second channel,and may be processed to monitor this first channel without the monitorsignal being corrupted by the second channel incident on the same pixelblock. Similarly for monitoring the second channel at a second receivingelement.

It is possible to perform the invention with embodiments using eithermultiphase or binary phase SLMs. Multiphase pixellated SLMs mayincorporate an integral or non-integral quarter-wave plate or a waveplate having a similar effect to provide polarisation insensitivitywhere a liquid crystal having out of plane tilt is used.

In a family of embodiments the SLM is treated as divided into blocks,with each block associated with a respective receiving element anddisplaying a respective hologram to route light to the receivingelement.

Although the aforegoing description discusses Fourier systems, theinvention is equally applicable to Fresnel systems which are likewiselinear and for which the field generated by the flashing pixel may beconsidered slowly varying across the receiving element. In practice thismeans that the field generated by the flashing pixel across thereceiving element varies as slowly or more slowly than the field createdat the receiving element by the original hologram, H₀(u,v).

The above description refers to LCOS SLMs. However the invention is notso limited but instead extends to the full scope of the appended claims.

1. A method of measuring amplitude and phase variations in a spatiallycoherent beam of light comprising causing the beam to be incident upon aspatial array displaying a pixellated first phase distribution, in ameasuring region of said spatial array, causing the phase distributionto change to a new value while retaining the first phase distributionoutside the measuring region, in the Fourier plane, determining thechange in intensity resulting from the change in phase distribution. 2.A method of characterising a spatially coherent beam of light,comprising disposing a LCOS SLM in the path of the beam; causing theLCOS SLM to display a first hologram pattern; at a location in said beamwhere the amplitude and phase of the beam are to be characterised,changing the hologram pattern to a second hologram pattern; andmeasuring the effect of said change by determining an intensity change.3. A method as claimed in claim 2, wherein, the output from the SLM isdetected in the Fourier plane to detect the Fourier output.
 4. A methodas claimed in claim 1, comprising measuring the intensity in a region ofthe Fourier plane where the amplitude distribution associated with thebeam modulated by the original hologram is relatively stronger, but theamplitude distribution at the Fourier plane of the field componentcreated by the perturbation in the hologram is relatively stronger,varying the position on the SLM where the perturbation is applied.
 5. Amethod as claimed in claim 4, further comprising taking the square rootof a set of values obtained.
 6. A method as claimed in claim 1,comprising stepping through a sequence of phase distributions.
 7. Amethod as claimed in claim 1, comprising varying the phase shift in arespective single pixel.
 8. A method as claimed in claim 3 comprisingmanipulating hologram patterns to obtain information related to acoherent coupling term to thereby derive amplitude information.
 9. Amethod as claimed in claim 3 comprising manipulating hologram patternsto obtain information related to a coherent coupling term to therebyderive phase information.
 10. A method as claimed in claim 1 wherein thestep of detecting is carried out at a single point.
 11. A method asclaimed in claim 1 wherein after changing to a new value, the phasedistribution in the measuring region returns to its original value. 12.Apparatus for measuring amplitude and phase variations in a spatiallycoherent beam of light, the apparatus comprising a pixellated spatialarray, each pixel being controllable to apply any of plural phase shiftsto input light, whereby the array displays a desired distribution ofphase modulation, means for causing the array to display a firstselected distribution of phase modulation; means for changing the firstdistribution in a measuring region of said spatial array to assume a newdistribution while retaining the first phase distribution outside themeasuring region, means disposed in the Fourier plane for determining achange in intensity of light resulting from the change in phasedistribution.
 13. Apparatus as claimed in claim 12, wherein the spatialarray has only two possible values of phase shift per pixel. 14.Apparatus as claimed in claim 12, wherein the spatial array has morethan two possible values of phase shift per pixel.
 15. Apparatus forcharacterising a spatially coherent beam of light, comprising a LCOS SLMarranged so that a said beam of light can be incident upon it; means forcausing the LCOS SLM to display a first hologram pattern; means forchanging the hologram pattern to a second hologram pattern at a locationin said beam where the amplitude and phase of the beam are to becharacterised; and means for measuring an intensity of light todetermine the effect of said change of hologram pattern.
 16. Apparatusas claimed in claim 15, wherein the means for measuring is disposed inthe Fourier plane to detect the Fourier output.
 17. Apparatus as claimedin claim 15, further comprising a lens for providing the Fourier output.18. Apparatus as claimed in claim 15, further comprising a mirror forproviding the Fourier output.